1. Introduction

Non-Hermitian Hamiltonian with PT-symmetry has received great attentions in recent years since it exhibits entirely real spectrum [1]. Depending on the degree of non-Hermiticity, the system may encounter a phase transition from the PT-symmetric phase with a real spectrum to the broken-PT phase with a complex spectrum. The phase transition point, also known as the exceptional point is a topological singularity in the parameter space, where eigenvectors (along with their eigenvalues) coalesce [2–4]. Spectral singularity is another type of singularity associated with non-Hermitian Hamiltonian [5]. It generally manifests itself as ultra-high transmission and reflection with vanishing bandwidth. Within the scattering formalism, such singularity corresponds to lasing threshold of cavity with gain. Exceptional point is a kind of singularity generally associated with a discrete spectrum, while spectral singularity is the singularity for a continuous spectrum. In optical systems, PT-symmetric complex potential in quantum mechanics is mapped into a complex permittivity satisfying $\epsilon (r) = \epsilon ^{*} (-r)$. PT symmetry and non-Hermitian photonics open new possibilities by controlling the imaginary part of the dielectric permittivity, and by considering gain and loss on equal footing. In most of the early works in optics system, PT-symmetric structures are constructed by parallel waveguides with alternating gain and loss either along or across the propagation direction. A wide range of novel electromagnetic phenomena, such as power oscillation [2], loss-induced transparency [4], asymmetric light propagation [6–11], coherent perfect absorber-laser [12,13] and non-reciprocity [14,15] are demonstrated in waveguide configuration.

Besides the well studied PT-symmetric waveguide structure, metamaterials structure is also a promising platform for studying non-Hermitian Hamiltonian, since it offers preselected resonant elements with precise control over the structural parameters, which corresponds to predetermined smart optical properties and preset controllable coupling [16–20].

By exploring the interplay between introduced gain and intrinsic loss of components in the hybridized metamaterials, similar mappings to PT-symmetric Hamiltonians have been theoretically and experimentally explored including coherent perfect absorber-laser [21,22], loss-induced super scattering [23,24] and unidirectional phase singularity [25,26]. Unlike the waveguide configuration, where loss and gain are consider on equal footing, it is difficult to introduce gain as large as the total loss of a metallic nanostructure [21,27,28]. Hence, it is of great importance to lower the critical gain level for triggering the exceptional points or spectral singularity of active metamaterials systems. Feng et al. and Zhang et al. have demonstrated spectral singularity in active metamaterial systems, but at relatively high gain levels [23,24]. Most recently, Li et al. proposed an active epsilon-near-zero plasmonic waveguides structure to display non-Hermitian optical properties at very low gain level, taking advantage of the field concentration effect of this particular structure [28].

Here we investigate the relationship of coupling strength and critical gain level for triggering spectral singularity in an active hybridized metamaterials system (see Fig. 1(a)). With the help of transmission line combined with a lumped element (TLLE) model, we reduce the threshold of the gain level to trigger the singularity significantly, by varying the coupling coefficients between different metal strips in the proposed structure. Furthermore, by introducing small eigenfrequency mismatch between bright and dark modes, we demonstrate rapid switching between two spectral singularities at different frequencies. Ultra-high transmission and reflection at the same frequency at the spectral singularity is confirmed by a full-wave three-dimensional simulation. Our investigation provides new insight in designing non-Hermitian metamaterials system with novel optical properties. And the results provide feasibility for future applications in active metamaterials, such as low threshold lasing, switching, sensing and nonlinear optics.

 

Fig. 1. (a) Schematic diagram of the hybridized metamaterials comprising of a dipole nanoantenna and a quadrupole antenna. (b) The TLLE model corresponding to the proposed metamaterials structure shown in Fig. 1(a).

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2. TLLE model of the hybridized metamaterials

The hybridized metamaterial under investigation is shown in Fig. 1(a). Here, we study a fundamental configuration, i.e., a dipole bright mode coupled with a quadrupole dark mode. Metal strip 1 (bright mode) supports a localized SPP resonance that can be excited through the x-polarized incident plane wave directly. A dark mode with nearly the same resonant frequency is supported by strip 2 and 3, resonating antiparallelly. It can only be excited by the near field coupling from the bright mode since strip 2 and 3 are perpendicular to the incident electric field. To eliminate the influence of unnecessary factors, the metamaterials structure is set in the vacuum. Similar system has been previously considered to achieve electromagnetically induced transparency (EIT) [29]. Here we incorporate gain in the dark mode to probe non-Hermitian optical properties of such hybridized metamaterials system. As we have suggested in introduction, unlike the PT-symmetry waveguides or on-chip resonators structure [4,14], the resonance of metamaterials structure undergoes large scattering loss in addition to its intrinsic absorption loss. And the scattering loss of the proposed sub-wavelength structure is much bigger than its absorption loss [21]. Although the combination of metamaterials with gain is increasingly mastered in recent years, it is still difficult to introduce such big amount of gain that equals the total loss. In order to better understand the singularity associated with non-Hermitian sub-wavelength systems and to manipulate their functioning gain level, we revisit the TLLE models we utilized before to describe this active metamaterials structure [30,31].

Compared to the widely used Lorentz oscillator model, the parameters of the TLLE-model (resistance $R$, inductance $L$, capacitance $C$) are more intuitive because they correspond physically to the properties of the metamaterials and plasmonic structures. By treating all the resonators as a whole metafilm and combining it with transmission line theory, circuit models with lumped element parameters have been successfully used to analyze metamaterials working at infrared region [30]. Figure 1(b) shows the TLLE-model for the metamaterials structure. In this model, the transverse electromagnetic wave propagates through free space with wave impedances $Z_i$ and $Z_o$ ($Z_i=Z_o=Z_0$, $Z_0$ is the free space intrinsic impedance). The metamaterials structure is described by two coupled, lumped-element RLC oscillators behaving together as a boundary with complex impedance. $R_1$, $L_1$ and $C_1$ represent the lumped element parameters of the first metal strip, while $R_2$, $L_2$ and $C_2$ represent that of the oscillation of strip 2 and 3 together. In near field regime, the coupling between bright and dark modes is specified by mutual magnetic inductance $M$. Because of the orthogonality of the two modes, only the magnetic field coupling is took into account. $M=\kappa \sqrt {L_1L_2}$, $\kappa$ is the coupling coefficient between two inductors in the range from 0 to 1. The gain introduced in the dark mode is considered as a negative resistance $-R_g$ in circuit of the dark mode, acting as a current source [32]. The magnitude of $-R_g$ is independent of the induced current in the dark mode and it is proportional to the amount of introduced gain.

The proposed metamaterials configuration is a two-port network. In this case, we can use the characteristic impedance $Z_{m}$ of the metamaterials structure to construct the 2 $\times$ 2 ABCD matrix, which can relate the total electric and magnetic fields at the output and input ports of this system. It takes the form [33]

It can be rigorously shown within semiclassical laser theory that the first lasing mode in any cavity is an eigenvector of the electromagnetic S matrix with an infinite eigenvalue [13]. Such diverging eigenvalue is also known as the spectral singularity of the system [5]. And with Eq. (2) and (1), we are able to analyze the impedance matching conditions where the lasing frequency points occur. In order to achieve simultaneously diverging reflection and transmission coefficients (lasing or superscattering), the denominators of $S_{11}$ and $S_{12}$ need to become equal to zero. In fact, all elements in the scattering matrix diverge at the spectral singularity. Therefore, by letting $AZ_o+B+(CZ_o+D)Z_i$ equals zero, the real and imaginary parts of the characteristic impedance of the proposed metamaterials structure should satisfy the following lasing conditions:

Without the effect of introduced gain, the real part of $Z_m$ is always positive at real frequencies, indicating loss in the transmission line. Yet the introduced gain in the metamaterials component can bring real negative part to $Z_m$, indicating amplification of waves in the line. And once the effect of gain has brought the characteristic impedance down to $-Z_0/2$, laser action will happen at a real frequency [28]. We will show below that the critical gain level for satisfying Eq. (3) is crucially related to coupling strengths between different strips in the system.

3. Simulation details

Finite element analysis solver Comsol Mutiphysics is used to obtain the scattering spectra of the metamaterials under all conditions. The geometry of the metamaterials structure is $l_1=128\ \mathrm {nm},\ l_2=100\ \mathrm {nm},\ w_1=50\ \mathrm {nm},\ w_2=30\ \mathrm {nm},\ t=20\ \mathrm {nm},\ s=30\ \mathrm {nm}$, and $h=50\ \mathrm {nm}$. The distances between different strips, $h$ and $s$, are varied in the simulations. Silver is selected as the material due to its low intrinsic loss at the interested frequency range. During the simulations, one unit cell of the metamaterials was placed inside a simulation region with the transverse boundary conditions set to be periodic. Perfectly matched layers (PML) are applied at longitude boundaries to eliminate any reflection at the boundaries. The permittivity of silver ($\epsilon =\epsilon '+i\epsilon ''$) is described by the Drude model of $\epsilon =\epsilon _\infty -\omega _p^2/(\omega ^2+i\gamma _p\omega )$, where $\omega _p=2.1961\times 10^{15}\ \mathrm {Hz}, \gamma _p=4.3439\times 10^{12}\ \mathrm {Hz}, \ \mathrm {and}\ \epsilon _\infty =3.7$ [21].

The effect of gain is crucial for bringing the spectral singularity of non-Hermitian systems to a real frequency [13]. Methods include incorporating optical active (gain) materials or doping quantum dots or organic dyes in a host dielectric material can be utilized to introduce gain in metamaterials structures [21,35,36]. Here we decide not to focus on detail realization of these methods, because our goal here is to present a practical way to manipulate the critical gain level of non-Hermitian optical phenomena. we assume that the gain effect reverses the sign of the imaginary part of permittivity $\epsilon ''$ and model the permittivity of the dark mode by $\epsilon ' + i(-\epsilon _g)$ [24]. The magnitude of $-\epsilon _g$ of the dark mode is proportional to the amount of gain in it. We also ignore the frequency dispersion of the gain medium, assuming that the $-\epsilon _g$ is a constant over the spectrum region within our interest.

4. Results and discussion

4.1 Spectral singularity at low gain level

The hybridized metamaterials structure can be regarded as a ’two-particle’ system, as the bright and dark modes are represented by two sets of RLC elements in the TLLE model. The introduced gain in the metamaterials is modeled by a negative resistance $-R_g$ connecting to the RLC circuit. The impedance of hybridized metamaterials in this case is written as

 

Fig. 2. (a) The relationship between gain level $-\epsilon _g$ and the normalized resistance $-\tilde {R}_g$. (b) The relationship between coupling coefficient $\kappa$ and distance between bright and dark modes $h$. (c) The normalized resistance $-\tilde {R}_g$ at spectral singularity as a function of $h$. Color dots on the curve indicate different $h$ under study.

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The resonance frequencies of the bright and dark modes are both 440 THz under the current geometry parameters, suggesting $L_1 \times C_1 = L_2 \times C_2$. Note that the imaginary parts of $X_1$ and $X_2$ vanish at the resonance frequency. So the second part of the lasing condition in Eq. (3) is naturally satisfied at 440 THz and the characteristic impedance $Z_m$ is reduced to $R_1-M^2\omega _0^2/ \tilde {R}_g$, $\omega _0= 2\pi \times 440\ \mathrm {THz}$. Now the characteristic impedance becomes a real number and is a function of the ratio of the square of coupling strength $M$ and the normalized resistance $-\tilde {R}_g$. Therefore, if we set a relatively small coupling strength $M$, we will only need a small $\tilde {R}_g$ to satisfied the first part of lasing condition. We also know from Fig. 2(a) that $-\tilde {R}_g$ is linearly proportional to the gain level $-\epsilon _g$. In another word, the coupling strength between the bright and dark modes can be varied to manipulate the critical gain level of spectral singularity in the hybridized metamaterials system. In a near field limit, the coupling strength $M$ decreases with increasing $h$, the distance between bright and dark modes, as is shown in Fig. 2(b). So we substitute the fitted curve of $h$ and $\kappa$ into Eq. (4) and calculated the relationship between $h$ and critical value of $-\tilde {R}_g$ that satisfied the lasing condition. One can see in Fig. 2(c) that the normalized resistance $-\tilde {R}_g$ at the spectral singularity decreases parabolically with the distance between the bright and dark modes $h$ as expected. So we conclude that $h$ can be increased to reduce the critical gain level.

We simulate the transmission and reflection spectra at two different $h$, 75 nm and 50 nm, which are indicated by color dots in Fig. 2(c). The simulated output intensity is defined as $\mathrm {log}_{10} |r^2+t^2|$. The simulated output intensity spectra versus the imaginary part of permittivity ($-\epsilon _g$) and frequency at different $h$ are displayed in Figs. 3(a) and (c). Figures 3(b) and (d) show the corresponding calculated results by the TLLE model. The simulation results are in good agreement with the analytic results from the TLLE model. One can see that there is an ultra-high peak of $|r^2+t^2|$ at 440 THz in each color map. And apart from each critical gain level, such a giant transmission and reflection disappear abruptly. The critical gain level has dropped from $-\epsilon _g=1.26$ to 0.27 when $h$ is increased from 50 nm to 75 nm. The simulation results of the output intensity spectra when $-\epsilon _g=1.26$ and 0.27 are shown as red lines in Figs. 3(e) and (f), while the corresponding analytical results from TLLE model are shown in blue lines with circles for comparison. The gain level in the simulations ($-\epsilon _g=1.26$ and 0.27) matches well with the normalized resistance in the model ($-\tilde {R}_g=194$ and 49).

 

Fig. 3. Simulated output intensity $\mathrm {log}_{10} |r^2+t^2|$ versus the imaginary part of permittivity ($-\epsilon _g$) and frequency at $h=75\ \mathrm {nm}$ (a) and at $h=50\ \mathrm {nm}$ (c). Corresponding calculated output intensity at $h=75\ \mathrm {nm}$ (b) and $h=50\ \mathrm {nm}$ (d). The simulation results (red lines) and the corresponding analytical results (blue lines with circles) from TLLE mode of the output intensity $\mathrm {log}_{10} |r^2+t^2|$ at the spectral singularity when $h=75\ \mathrm {nm}$ (e) and $h=50\ \mathrm {nm}$ (f), respectively.

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Besides the coupling strength between the bright and dark modes, coupling strength inside the dark mode (between strip 2 and 3) can also be varied to reduce the critical gain level. As illustrated in Fig. 4, we separate the dark mode into two different resonators in the TLLE model. Now $R'_2, L'_2, C'_2$ and $R'_3, L'_3, C'_3$ represent resonances of the second and third metal strips. In near field regime, the coupling between bright and dark modes is specified by mutual magnetic inductance $M_{12}$ and $M_{13}$, while $M_{23}$ stands for coupling between $L'_2$ and $L'_3$. And we have $M_{12}=M_{13}=\kappa _{12}\sqrt {L_1L'_2}=\kappa _{13}\sqrt {L_1L'_3}$ and $M_{22}=\kappa _{23}\sqrt {L'_2L'_3}$. The characteristic impedance of the hybridized metamaterials is now given by

 

Fig. 4. The TLLE model corresponding to the proposed metamaterials structure when the dark mode is represented separately.

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Again, the coupling coefficient between strip 2 and 3 can be modulated by displacement $s$ between these two strips, as indicated in Fig. 5(a). We substitute the fitted curve of $s$ and $M_{23}$ into Eq. (5) and calculate the relationship between the normalized resistance $-\tilde {R}_g$ at spectral singularity and $s$ when $h=75 \mathrm {nm}$. Compared with Fig. 2(c), the relationship between $s$ and the critical gain level is not that straight forward, as one can see in Fig. 5(b). But since it is a concave function, we can reduce the critical gain level by decreasing or increasing $s$ from 30 nm, which corresponds the maximum value of $-\tilde {R}_g$. Notice that $s=30\ \mathrm {nm}$ is where the resonance frequencies of the bright and dark modes coincide. Because of the bigger slope of the curve in the left side, we narrow the gap between strip 2 and 3 to further reduce the critical gain level.

 

Fig. 5. (a) The relationship between coupling coefficient $\kappa _{23}$ and distance $s$ between strip 2 and 3. (b) The normalized resistance $-\tilde {R}_g$ at spectral singularity as a function of $s$. Color dots on the curve indicate different $s$ under study.

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Figure 6 depicts the simulated and calculated output intensity as a function of the imaginary part of permittivity ($-\epsilon _g$) and frequency when $h=75\ \mathrm {nm}$, $s=20\ \mathrm {nm}$, which is indicated by the green dot in Fig. 5(b). Compared with Fig. 3(a) where $h=75\ \mathrm {nm}$, $s=30\ \mathrm {nm}$, one can see from Fig. 6(a) that after narrowing the gap between strip 2 and 3, the critical gain level of spectral singularity is reduced from $-\epsilon _g=0.27$ to 0.05. The lasing frequency is now 419 THz because the eigenfrequency of the dark mode is red shifted form that of the bright mode (440 THz), when $s$ is narrowed from 30 nm to 20 nm. The simualtion results are in good agreement with the calculated results from TLLE model and the critical gain level $-\epsilon _g$ in simulation matches well with the normalized resistance $-\tilde {R}_g$ at the singularity.

 

Fig. 6. (a) Simulated output intensity $\mathrm {log}_{10} |r^2+t^2|$ versus the imaginary part of permittivity ($-\epsilon _g$) and frequency when $h=75\ \mathrm {nm}$, $s=20\ \mathrm {nm}$. And its corresponding analytical results (b).

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Basically, the imaginary part of permittivity of the dark mode $-\epsilon _g$ at singularity when $h=75\ \mathrm {nm},\ s=20\ \mathrm {nm}$ (0.05) is more than 20 times smaller than that when $h=50\ \mathrm {nm},\ s=30\ \mathrm {nm}$ (1.26). So far, we study a minimal design of hybridized metamaterials structure and realize non-Hermitian optical properties at very low gain level, which is a significant improvement over the work in Ref. [23] and [24] in terms of the functioning gain level.

It is worth mentioning that a single dark resonant element whose radiative decay rate, even if small, always remains finite and therefore can still be directly excited by an external field. With the TLLE model and Eq. (3), one can easily find that the critical gain level of spectral singularity in such system will also be very low. Actually, in the proposed structure, the radiative bright mode can be viewed as a continuum state for the dark mode. For a dark mode, coupling with the bright mode corresponds to radiating to the continuum state (external field). Therefore, decreasing the coupling strength is equivalent to lowering the radiative decay rate of the dark mode. Theoretically, these two configurations are basically equivalent in terms of achieving low gain level for spectral singularity. But we believe that there are two main advantages of the coupled modes configuration:

1. From a practical stand point, the coupled bright-dark scheme can be excited by normal incident plane wave. Furthermore, the coupling coefficient can be directly controlled by the displacement between different components. While for a single dark mode, oblique incidence may need to be utilized and the control of the radiative decay rate of the dark mode may be difficult.

2. The spectral singularity in a coupled modes configuration is a ’high-order’ spectral singularity. Ref. [24] suggests that the spectral singularity in a coupled modes configuration is a hybridized one concerning more than one modes. We envision that such ’high-order’ spectral singularity may promise future applications besides lasing, like the high-order exceptional point which has been recently found to have great potential in sensing [20].

4.2 Rapid switching between two different spectral singularities

In the above analysis, the coupling coefficient between strip 2 and 3 are varied to lower the critical gain level. Notice that when the displacement $s$ between strip 2 and 3 is varying, the eigenfrequency of the dark mode is shifting from that of the bright mode (440 THz). In this case, according to lasing condition in Eq. (3), more than one set of gain level and frequency are supposed to satisfied Eq. (3), which means that two or more spectral singularities at different frequency can be accessed at different gain level. We substitute Eq. (5) into Eq. (3) and get

One can see that it is a pair of coupled high order polynomial equations. All spectral singularities in the metamaterials system can be obtained directly by solving Eqs. (6) and (7), but this process is cumbersome [37]. Here, we numerically calculate the output intensity using Eqs. (2) and (5) by taking a series of discrete value of $\kappa _{12}$ and $\kappa _{23}$ (with a step of 0.001). Scattering matrix of the proposed system is calculated at all frequency and gain level to determine the spectral singularity in the system.

It is found from the calculation that when the coupling strength between the bright and dark modes is above certain threshold, more than one spectral singularity can be achieved at different frequency. Meanwhile, the coupling coefficient $\kappa _{23}$ can be varied to adjust the gain level difference between two singularities at different frequency. At $\kappa =\sqrt {2}\kappa _{12}=0.135$ and $\kappa _{23}=0.11$, we find that two spectral singularities at different frequency can be achieved and their gain level difference is small. Accordingly, we rearrange the displacement between different strips and set $s=33 \ \mathrm {nm}$, introducing small mismatch between the eigenfrequencies of bright and dark modes. We also set $h=20 \ \mathrm {nm}$, coupling these two modes strongly. These geometry parameters correspond to the TLLE model parameters $\kappa _{12}$ and $\kappa _{23}$.

The contour of the simulated output intensity is shown in Fig. 7(a). Two peaks resembling two different spectral singularities lie in the color map. The first one locates at 412 THz when the gain level in the dark mode is $-\epsilon _g = 1.73$. And when we increase the gain level to $-\epsilon _g=1.85$, the original spectral singularity will evolve into another one at a very different frequency (464 THz). The two singularities locate at 415 THz, $-\tilde {R}_g=497$ and 467 THz, $-\tilde {R}_g=534$ in the analytical results, showing good agreement with the simulation results. We believe that the fitting process of obtaining the TLLE parameters leads to the small discrepancy in the frequency of spectral singularities. Laser action at different frequencies can be achieved in the same structure and can be switched between each other by varying small amount of gain. Such exotic property can be utilized in sensing and lasing areas. Figures 7(b) and (d) depicts the simulated and calculated transmission spectra at the two singularities. One can see that besides each main peak, there is a small side peak at the frequency of the other singularity. Such small overlap of the two singularities is acceptable since the side peak intensity is more than a thousand times small than the main peak.

 

Fig. 7. (a) Color map of the simulated output intensity $\mathrm {log}_{10} |r^2+t^2|$ versus the imaginary part of permittivity ($-\epsilon _g$) and frequency when $s=33 \ \mathrm {nm}$ and $h=20 \ \mathrm {nm}$. (c) Corresponding calculated output intensity at $s=33 \ \mathrm {nm}$ and $h=20 \ \mathrm {nm}$. Simulated (b) and calculated (d) transmission spectra of the system at the two singularities.

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It is worth mentioning that the threshold value of the coupling coefficient between the bright and dark modes is $\kappa =0.075$, when $s=33\ \mathrm {nm}$. It is found from the numerical calculation that only one spectral singularity can be achieved at all gain level when the coupling coefficient between the bright and dark modes $\kappa$ is smaller than 0.075. This can be understood that because of the pure near field coupling between the bright and dark modes, when the coupling strength is below the threshold, the energy exchange between these two modes are limited, so the gain effect in the dark mode can only bring its own eigenmode to the spectral singularity. When the coupling strength is above the threshold, these two modes interplay strongly, so the gain in the dark mode can bring both two eignmodes to their spectral singularities. Therefore, the functioning gain level in this case is rather high, which may limit its practical application. However, methods such as incorporating epsilon-near-zero substrate which enabling field concentration could be utilized here to lower its functioning gain level [28]. Moreover, the design of rapid switching between two spectral singularities could be transferred into on-chip configurations (waveguide or micro-resonator), where the critical gain level of spectral singularity is much lower [5,37].

In Ref. [5,37–39], A. Mostafazadeh firstly provide an physical interpretation for the spectral singularity. Systems with complex potential, i.e. waveguide configurations where a certain segment is filled with gain medium, are investigated. Such waveguide systems support continuous state above their cut-off frequency. The lasing condition in these structures is a pair of coupled real transcendental equations (with $\tan ^{-1}$). There is a discrete set of frequencies that satisfy these equations, each represents a certain mode of singularities [37]. While in the proposed metamaterials system, the lasing condition is now a pair of coupled polynomial equations (substituting Eqs. (4) or (5) into Eq. (3)). Due to resonant nature of the metamaterial structure, the number of spectral singularities of the system is limited by the number of resonant modes in the system.

4.3 Introduction of gain materials

In the above simulations, we follow the treatment of material gain in Ref. [19,24], varying the imaginary part of the permittivity ($-\epsilon _g$) of silver strips in the dark mode directly to model the gain level. It is a convenient way to model the gain effect in simulation but not the practical way to introduce gain materials. Consequently, we simulate a practical implementation of the gain-assisted hybridized metamaterials as shown in Fig. 8(a). The proposed structure is fabricated on a glass substrate and is embedded in a polymethyl methacrylate (PMMA) layer with height of 60 nm. The red box with dimension $l_2=120\ \textrm {nm}$, $w_2=100\ \textrm {nm}$ and $t_2=60\ \textrm {nm}$ is centered at the dark mode and is doped with organic dyes or quantum dots to introduce gain effect. The proposed hybridized metamaterials is set at the optimized geometry parameters, $h=75\ \mathrm {nm},\ s=20\ \mathrm {nm}$. The refractive index of glass and PMMA is 1.5 and 1.49, respectively [27], while the refractive index of the doped region is $1.49-ik$. $k$ is varied to model the gain level in the dark mode.

 

Fig. 8. (a) Schematic of the practical implementation of the gain-assisted hybridized metamaterials. (b) Color map of the simulated transmission $\mathrm {log}_{10} |t^2|$ versus the imaginary part of refractive index of the doped region ($k$) and frequency.

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Figure 8(b) shows the simulated transmission $\mathrm {log}_{10} |t^2|$ versus $k$ and frequency. One can see that the spectral singularity locates at 291.5 THz and $k=0.0144$. Compared with Fig. 6, the frequency is red shifted due to the increase of the permittivity of the surrounding medium. A more conventional form of gain level for laser physics can be calculated as $\chi = 4\pi k/\lambda = 1760\ \textrm {cm}^{-1}$ [40]. Such a gain level at about 1000 nm is attainable by utilizing either organic dyes [41] or semiconductor materials [42].

5. Conclusion

In conclusion, by using a transmission line model, we studied the optical properties of a gain assisted planar metamaterials structure. The TLLE model gave us a deep insight into the physical mechanism of the spectral singularity in an active metamaterials system. It is understood that the critical gain level of spectral singularity in coupled metamaterials structures is crucially related to the coupling coefficients between different components. By optimizing these coupling coefficients, we demonstrated the spectral singularity of the proposed structure at very low gain level. Furthermore, by introducing small difference between eigenfrequencies of the bright and dark modes, we demonstrated two spectral singularities at different frequency in the same structure which can be rapidly switched between each other. The simulation results are in good agreement with the analytic results from the TLLE model. The demonstrated non-Hermitian optical properties will apparently enlighten future PT-symmetry metamaterials design and may bring active metamaterials structure a step further to practical implementation.